On Fuzzy (p, α, pα)-compact subspaces

In this paper, we study the relationships among fuzzy (compact, pre-compact, α-compact and pα-compact) subspaces; also we will discuss the relationships between fuzzy (compact, p-compact, α-compact and pα-compact) spaces and fuzzy (compact, precompact, α-compact and pα-compact) subspaces.


Introduction:
The fuzzy issue has entered nearly all mathematics branches, fuzzy set (FS) has membership degree elements and such sets issue was widespread 1 st via Professor Lotfi A. Zada [1] in 1965. In 1968, Chang [2] announced the fuzzy topological Desscription of spaces and stretched in a straightforward fashion few crisp topological spaces issues to spaces as fuzzy topologically. The fuzzy topology (FT) was initiating in 1968 along article of Chang [2], as well might be regarded as a new mathematics branch, so several extra structures were considered via utilizing (FS)s and the correlated problems in applied and pure mathematics. In 1974, Wong debated and generalized few spaces properties of fuzzy topologically. In 1980, Ming and Ming utilized (FT) to describe the neighborhood fuzzy point (FP) structure. In 1991, Shahna described the -open and pre-open issue in fuzzy topological space (FTC) [3]. Rubasri and Palanisamy are reviewing the (FSs) of p-open (2017) [4]. [5, P.211-220], [6, P. 137-150] A (FP) in is (FS) described as following:

Desscription
and belong to so ∧ ∈ , belong to for each ∈ , so so does ⋁ ∈ When is a (FT) on , so the pair ( , ) is termed a (FTC  (1) If be a (FOS) in a fts X, so ≤°.
assumed to be p − (quasi − neighborhood) system and signified via .

Desscription
is assumed to be p − (quasi − neighborhood) system and signified via .

Desscription
(FP) is assumed to be  − (quasi − neighborhood) system and signified via .

Desscription
Suppose be a (FS) in a ( ). So: i-fuzzy p − interior, signified via °i s all p-open subsets union that is enclosed in .
ii-fuzzy p − closure , signified via is all fuzzy p-closed subset intersection comprises . iii-fuzzy  − interior , signified via ° is all -open subsets union that are enclosed in .
iv-fuzzy  − closure , signified via is all fuzzy -closed subset intersection comprises . v-fuzzy p − interior , signified via ° is all p-open subsets union that are enclosed in .
vi-fuzzy p − closure , signified via is all fuzzy p-closed subset intersection comprises .

Comment
Suppose , are 2 (FSs) in a ( ), so:  is a fuzzy closed iff = ;

6)
is a fuzzy p-closed iff = ; iii) It is straightforward.

Remark
In  .  [8, P.131-139] (FSs) family has property as fixed intersection when and only when the members intersection of every finite sub-family is as not-empty. Fuzzy (pα, α, p) compact subspaces 2.1. Desscription [10] X as fuzzy space A is termed fuzzy compact when each (FO) of cover X of sub-cover being finite.

Desscription
(FSs) family in a (FTC)( , ) is assumed to be a fuzzy p-open (FS) cover when and only when ≤ ⋁{ : ∈ } and every member is p-open (FS). Sub-cover is a sub-family that is covered as well. . Sub-cover is a sub-family that is covered as well.

Desscription
(FSs) family in a (FTC)( , ) is assumed to be a fuzzy p-open cover when and only when ≤ ⋁{ : ∈ } and every member is p-open (FS). Sub-cover is a sub-family that is covered as well.

Desscription
as fuzzy space is termed fuzzy p-compact when every cover fuzzy p-open of sub-cover being finite.

Desscription
as fuzzy space is termed fuzzy -compact when every cover fuzzy -open of sub-cover being finite.

Desscription
as fuzzy space is termed fuzzy p-compact when every cover fuzzy p-open of sub-cover being finite.